Math before words: What babies know about numbers

Back in the 20th century, Swiss psychologist Jean Piaget would have said they know nothing at all.

But today’s cognitive scientists have overturned the old ideas.

New experimental research reveals a fascinating new world of baby cognition, one in which babies can

• recognize the approximate difference between two numbers

• keep precise track of small numbers, and

• do simple subtraction and addition problems.

Moreover, when babies perform these feats they activate the same parts of the brain that are associated with mathematical thinking in adults.

Here I review the latest discoveries. I begin with what babies
know about numbers—big and small—and end with a discussion of baby
“arithmetic” and the brain.

What babies know about numerosity

Psychologists define “numerosity” as the number of things in a set.

Although we can get a precise measure of numerosity by counting,
it’s possible to appreciate numerosity in a more approximate way.

For instance, if I ask you to glance in a crowded elevator and
estimate how many people are there, you can probably make a pretty good
guess.

Glance inside two elevators, and you’ll also be pretty good at guessing which one has more people in it.

All without counting.

It turns out that adult humans aren’t the only creatures who can
pull this off. A variety of non-verbal creatures--including monkeys,
rats, fish, and human infants--can detect the approximate difference in
magnitude between two sets (Dehaene 1999; Piffer et al 2012).

Show them two displays—one featuring 8 items and the other only 4
items—and they will respond differently depending on what they see.

And the difference is observable in the brain.

When 3-month old infants were shown a series of displays
depicting different quantities of objects, the electrical activity of
their brains changed in response (Izard et al 2008).

For example, babies were presented with a continuous stream of
images, each depicting 4 objects. From image to image, the objects were
arranged differently. But the total number of objects remained the same (see the first four slides in the image below).

Infants subjected to this program showed signs of boredom. But their brain activity (as measured by event-related potentials, or ERPs) would perk up if they were next presented with an image showing 8 objects (see the fifth, "test" slide in the image above).

What’s more, the part of the brain that was activated—the parietoprefrontal network—was the same region that lights up when adults process information about numbers (Izard et al 2008).

A strong case for “baby number sense”

Were the babies really responding to the change in numerosity? It certainly seems so.

Babies reacted similarly when presented with pairs of small numbers (2 vs 3) and distant, large numbers (4 vs 12).

And researchers controlled for several, non-numerical variables
that could have influenced babies—like the total surface area of the
objects, the average area devoted to each object, the total luminance of
each image, and the total area occupied by each group of objects.

So babies weren’t simply attending to the continuous extent of “stuff” in each image.

In fact, other experiments suggest that babies pay more attention to change in number than they do to changes in continuous extent or surface area (Cordes and Brannon 2008).

Nor is it all “in the eye.” If babies’ feats of numerical
discrimination were confined to visible objects, we might suspect that
the ability is domain specific—a specialization of the visual system
rather than a more general conceptual ability. But babies aren’t
one-trick ponies. In addition to distinguishing visual arrays, babies
can also detect differences in the number of sounds and in the number of
actions (Lipton and Spelke 2003; Wood and Spelke 2005).

Babies can even understand ordinality--the idea that numbers can
be arranged in order of magnitude. When 11-month old babies were
presented with sequences of numerosities, they could discriminate
between sequences that increased and sequences that decreased (Brannon
2002).

Of course, this doesn’t mean that babies can distinguish any two numbers. There are limits. For instance:

• Six-month old babies have a hard time distinguishing two
quantities if the ratio between them is less than 2:1. In other words,
they can handle 8 vs. 16, but fail to distinguish 8 vs. 12 (Xu and
Arriaga 2007; Lipton and Spelke 2003).

• Nine- and 10-month olds can make finer distinctions (8 vs. 12).
But even these older babies don’t seem to discriminate between 8 and 10
(Xu and Arriaga 2004; Lipton and Spelke 2003).

• Babies don’t distinguish between increasing and decreasing
sequences of numbers until the approach the end of their first year.
When Elizabeth Brannon tested 9-month old babies, they failed the test
(Brannon 2002).

• Babies' have been stumped by experiments asking them to
distinguish a very small number from a larger one (e.g., 2 vs 4 or 2 vs
8). Kristy Van Marle (2013) thinks that's because babies rely on
different mechanisms to make judgments: A precise tracking mechanism for
numbers smaller than 4 and an approximate system for judging rough
magnitude. Babies don't know yet how to integrate the two data sets.

Nevertheless, the results are pretty impressive. When babies look
at a set of objects, they don’t just see a “bunch.” Babies have an
approximate sense of number.

Do babies act on what they know?

It seems that babies appreciate differences in magnitude. But can
babies put this knowledge to any use? Clever experiments suggest that
they can.

In one study, 14-month old babies watched experimenters place
toys in a box. Then experimenters asked the babies “What’s in the box?”
and let them search.

If a baby had seen only one toy placed in the box, he tended to
stop searching after he retrieved that ball. But if he’d seen two balls
go in, he’d search longer. Using this method, researchers reported that
babies could keep track of up to three items (Feigenson and Carey 2003).

In similar experiments, 21-month old babies watched while an
experimenter placed balls in a box one at a time (Langer et al 2003).
Then they were given these instructions:

“One at a time, with one hand, take all the balls out of the box…”

If babies had seen more than one ball placed in the box, they reached into the box more than once.

Experimenters also ran a trial in which babies saw the
experimenter place two balls in the box, and then pull one ball back
out. After watching these trials, babies who were asked to “take all the
balls out of the box” reached into the box only once.

Baby arithmetic?

These “balls in the box” experiments suggest that babies can perform other feats--namely, very basic calculations.

Placing two balls in a box--one after the other--amounts to an
exercise in addition (1+1). Similarly, removing one of two balls from a
box constitutes subtraction.

So babies who watch these scenarios—and form the correct inferences—are perhaps performing primitive math.

Or are they?

Some researchers have argued that babies don’t solve these
problems by doing addition or subtraction. Instead, they are relying on
an automatic system for tracking objects.

This system (which has been studied extensively in adults)
appears to track individual objects over time by opening a temporary
“file” for each one. The brain updates these files as objects move or
change (Kahneman et al 1992).

The system is limited by memory constraints—the average person
being able to track a maximum of 4 objects at once. But it’s an
effective system for coping with very small numbers.

According to the object file theory, babies detect the sudden
absence of an object because there is no longer a match between the set
of objects and the set of opened object “files.”

Similarly, the brain notices the unexpected appearance of a new object for which there is no file.

It’s not really arithmetic, then. More like a perception and memory task.

But even if babies use object files, we can’t rule out the
possibility that babies really do understand something about addition
and subtraction.

To resolve the question, we need to test babies’ knowledge of arithmetic without involving their object tracking systems.

And that’s exactly what Koleen McCrink and Karen Wynn set out to do.

What babies know about addition and subtraction

Cognitive scientists McCrink and Wynn suspected that babies really
could perform numerical computations--if only in a rough, approximate
way.

To test their idea, they presented 9-month old babies with
operations involving large number, numbers too large for the object
tracking system to handle (McCrink and Wynn 2005).

During the experiment, researchers showed babies several animated films depicting a group of moving rectangles.

In films presenting the addition scenario, 5 rectangles moved
around the screen and then hid behind an occluder. Next, 5 more
rectangles followed. The movies ended in one of two ways:

• the occluder moves off to reveal 10 rectangles (the correct outcome), or

• the occluder moves off to reveal 5 rectangles (the incorrect outcome)

The subtraction scenario was similar, except that it began with
10 rectangles that hid behind an occluder. Subsequently, 5 rectangles
slide back “out” and traveled—one by one—off the screen. The movies
ended with either

• the occluder moving off to reveal 5 rectangles (the correct outcome), or

• the occluder moving off to reveal 10 rectangles (the incorrect outcome).

In both scenarios, babies stared longer at the screen when it displayed the incorrect outcome.

The implications? Babies didn’t expect the wrong answers.

They weren’t looking longer because they preferred a certain
number of rectangles. The incorrect outcomes were associated with
different numbers (5 for the addition scenario, 10 for the subtraction
scenario).

So it seems that the babies understood two basic points—that
addition makes a set more numerous, and subtraction makes it less
numerous.

Wynn and McCrink concluded that these babies really do have
procedures for numerical calculation, procedures that don’t depend on
the automatic object-tracking system.

So how do babies do it?

It appears that babies--like adults and like many non-human
animals--can track very small sets of objects automatically. They also
have the capacity to compare large sets in an imprecise way.

The two feats are distinct and get processed in different parts
of the brain. A study measuring brain activity in 7-month infants
reports distinct neural signatures associated with each task (Hyde and
Spelke 2011).

And it appears there are specialized neurons for keeping track of approximate quantities greater than 3.

Experiments on monkeys have identified "accumulator neurons,"
brain cells that are activated by visual displays of objects. The more
objects on display, the more these special neurons respond (Roitman et
al 2007).

The monkey accumulator neurons were found in the same part of the
brain that is associated with number-processing in humans. This
region--the lateral surface of the parietal cortex--has also been linked
with the ability to locate objects in space and time.

So when your baby looks at a large array of objects, it seems
likely that specialized accumulator neurons are firing off in her
parietal cortex. The neurons respond in a graded manner, and the extent
of their activity supplies her brain with an approximate representation
of number.

And the result? A baby that knows the difference between 4 and 8 cookies. Even before she knows how to ask for them by name.